Introduction

This document aims to produce a quantitative framework for the Lilborn reinterpretation of the cosmological redshift in regards to the Law of Universal Hydrogen Emission and the full spectral shift observed in redshifted galaxies.

The standard model provides a precise formula linking the redshift parameter z to the velocity of a receding galaxy and to the scale factor of the expanding universe. The Lilborn Framework, which redefines z as a measure of angular coherence strain (Δθ_strain), must now produce its own predictive equation that can be measured and tested.

Redefining z Structurally

In the standard model, redshift z is defined as:

z = (λ_observed – λ_emitted) / λ_emitted

This measures the fractional increase in wavelength. In the Lilborn Framework, this is reinterpreted as a change in resolved frequency due to angular misalignment, not due to motion.

Therefore, we propose:

z = k · Δθ_strain

Where k is a scaling constant derived from the geometry of the coherence field and Δθ_strain is the angular deviation from perfect alignment.

Functional Form of Δθ_strain

Δθ_strain is defined as the cumulative angular deviation between the source alignment and the observer’s local coherence field.

In first-order approximation, this could be modeled by:

Δθ_strain = θ_observer – θ_source

Where each θ is the angular resolution required for the Ӕ function to resolve the hydrogen emission line at a particular frequency.

This assumes that the hydrogen is present at both ends of the interaction and that its angular coherence threshold is known.

Proposed Equation

We now propose the following equation for structural redshift:

z = k · (θ_observer – θ_source)

Where k is determined empirically by matching known z-values to observed angular deviations. This structural definition creates a one-to-one mapping from angular geometry to spectral displacement without invoking velocity or time dilation.

Conclusion

This document fulfills the requirement by providing the first structural derivation of z. Future work will focus on calibrating the scaling constant k using high-precision astronomical data and exploring higher-order effects of field curvature and resonance density.

We now proceed with confidence into the predictive phase of the Lilborn Framework.

Produced by The Lilborn Equation Team:

Michael Lilborn-Williams

Daniel Thomas Rouse

Thomas Jackson Barnard

Audrey Williams